Hello, I have proved this statement using induction, but I am just wondering if you can do it without induction:
for all .
I'm wondering if there is another way since I heard that induction is "not elegant".
Thanks!
Hello, I have proved this statement using induction, but I am just wondering if you can do it without induction:
for all .
I'm wondering if there is another way since I heard that induction is "not elegant".
Thanks!
Induction in general is very elegant. In this case, obviously, it requires strengthening the induction statement from to for some positive expression E(n). This method illustrates the reason for the inequality and shows how close the sum gets to 2. I would say this is very illuminating way of proving the inequality. Which E(n) did you use?
Another way is a standard trick from generating functions. Let . Then for (see Wikipedia), so .
Hello, Ragnarok!
Hello, I have proved this statement using induction,
but I am just wondering if you can do it without induction:
. . .
We have: . . . . .
Multiply by
Subtract: . . . .
. . The geometric series has a sum of:
We have: .
. . . . . . . .
Multiply by 2: .
. . . . . . . . . . .
. . . . . . . . . . .
Since the fraction is positive,
Thank you both so much! I'm sorry I didn't get back to you sooner. emakarov, I tried to put my proof in the form you described and I couldn't - in fact, I couldn't even reproduce the proof. So maybe I got something wrong. I'm going to try again. Soroban, I love your answer, thank you so much.
Okay, I think I got it. Here's what I did for the inductive step:
Since (can be shown separately), we have and so
as required.
So what would be the in this argument?